Analyzing the quality of the expected value solution in stochastic programming

Abstract

Stochastic programs are usually hard to solve when applied to real-world problems; a common approach is to consider the simpler deterministic program in which random parameters are replaced by their expected values, with a loss in terms of quality of the solution. The Value of the Stochastic Solution -VSS -is normally used to measure the importance of using a stochastic model. But what if VSS is large, or expected to be large, but we cannot solve the relevant stochastic program? Shall we just give up? In this paper we investigate very simple methods for studying structural similarities and differences between the stochastic solution and its deterministic counterpart. The aim of the methods is to find out, even when VSS is large, if the deterministic solution carries useful information for the stochastic case. It turns out that a large VSS does not necessarily imply that the deterministic solution is useless for the stochastic setting. Measures of the structure and upgradeability of deterministic solution such as the loss using the skeleton solution and the loss of upgrading the deterministic solution will be introduced and basic inequalities in relation to the standard VSS are presented and tested on different case studies.

The multi-path Traveling Salesman Problem with stochastic travel costs

“…The multi-path Traveling Salesman Problem percentage error of the approximated optimum when compared to the mean of the expected value given by the Monte Carlo simulation (Maggioni and Wallace 2012). Table 1 reports the percentage gap for all combinations of the parameters, while varying the probability distribution (either Uniform or Gumbel) in Set1.…”

Stochastic versus Deterministic Approach to Coordinated Supply Chain Scheduling

“…Second, in this paper, a stochastic programming wait-andsee approach with its ability to handle uncertainty by probabilistic scenarios of disruption events is compared with a deterministic programming approach, in which the random parameters are replaced by their corresponding expected values to achieve the so-called expected value problem (e.g., Kall and Mayer [27]). The expected value problem is a MIP and is often used in practice as the related stochastic mixed integer program is in general much harder to solve, since it considers multiple scenarios (e.g., Durbach and Stewart [28] and Maggioni and Wallace [29]). The objective of both the wait-and-see approach and the expected value approach is to optimize expected performance of a supply chain under the two types of disruptions with respect to two conflicting objective functions, expected cost and expected service.…”

Prosumer bidding and scheduling in electricity markets

“…To decide whether a rolling horizon deterministic model is to run hourly or a few times a day would be able to capture enough of the dynamics and flexibility to compete with a stochastic model, a large-scale analysis should be performed over a long time horizon (Maggioni & Wallace, 2010;. We leave this to future research.…”